Final answer:
To create a matrix whose nullspace is spanned by the vectors (3,3,1,0) and (5,-1,0,1), you can use an augmented matrix approach with row reduction technique. The resulting matrix, A, is the negative transposition of the vectors that form the nullspace.
Step-by-step explanation:
The question involves creating a matrix whose nullspace is spanned by the vectors (3,3,1,0) and (5,-1,0,1). To determine such a matrix, we need to find a matrix that, when multiplied by these vectors, results in the null vector. Since the nullspace of a matrix A is the set of all solutions to the equation A×x = 0, we want A to satisfy A×(3,3,1,0)T = 0 and A×(5,-1,0,1)T = 0, where 0 is the null vector and T indicates the transpose operation.
One suitable method to construct such a matrix is to start with an augmented matrix that includes these vectors as the solutions to the homogeneous system and perform row operations until we obtain the reduced row-echelon form. The resulting augmented matrix will suggest a matrix whose nullspace contains the given vectors.
Let's start by setting up an augmented matrix where the vectors that span the nullspace are on the right side of the augmented matrix (corresponding to the null vector on the left side):
[ 0 0 0 0 | 3 5 ]
[ 0 0 0 0 | 3 -1 ]
[ 0 0 0 0 | 1 0 ]
[ 0 0 0 0 | 0 1 ]
Row reduction of this system won't change anything since all the row entries on the left are already zero. So, an appropriate A matrix would simply be the negative of the transposition of the vectors on the right:
A =
[ -3 -3 ]
[ -5 1 ]
[ -1 0 ]
[ 0 -1 ]